Separated presheaf

Idea

Definition

Recall that a sheaf on SS is a presheaf A∈PShSA \in PSh_S such that for all local isomorphisms Y→XY \to X the induced morphism PShS(X,A)→PShS(Y,A)PSh_S(X,A) \to PSh_S(Y,A) (under the hom-functorPShS(−,A)PSh_S(-,A)) is an isomorphism. (For an arbitrary class of morphisms VV, the corresponding condition is called being a local object.)It is sufficient to check this on the dense monomorphisms instead of all local isomorphisms. This is equivalent to checking covering sieves.

Definition

A presheafA∈PSh(S)A \in PSh(S) is called separated (or a monopresheaf) if for all local isomorphisms Y→XY \to X the induced morphism Hom(X,A)→Hom(Y,A)Hom(X,A) \to Hom(Y,A) is a monomorphism.

More generally, for a class VV of arrows in a category CC, an object A∈CA\in C is VV-separated if for all morphisms Y→XY\to X in VV, the induced morphism Hom(X,A)→Hom(Y,A)Hom(X,A)\to Hom(Y,A) is a monomorphism.

Remark

As for sheaves, it is sufficient to check the separation condition on the dense monomorphisms, hence on the sieves.

For {pi:Ui→U}\{p_i : U_i \to U\} a covering family of an object U∈SU \in S, the condition is that if a,b∈A(U)a,b \in A(U) are such that for all ii we have A(pi)(a)=A(pi)(b)A(p_i)(a) = A(p_i)(b) then already a=ba = b.

Definition

For A∈PShSA \in PSh_S the separaficationLsepAL_{sep}A of AA is the presheaf that assigns equivalence classes

LsepA:U↦A(U)/∼U,
L_{sep}A : U \mapsto A(U)/\sim_U
\,,

where ∼U\sim_U is the equivalence relation that relates two elements a∼ba \sim b iff there exists a covering{pi:Ui→U}\{p_i : U_i \to U\} such that A(pi)(a)=A(pi)(b)A(p_i)(a) = A(p_i)(b) for all ii.

This construction extends in the evident way to a functor

Lsep:PSh(S)→Sep(S).
L_{sep} : PSh(S) \to Sep(S)
\,.

Proposition

This functor LsepL_{sep} is indeed a left adjoint to the inclusion i:Sep(S)↪PSh(S)i : Sep(S) \hookrightarrow PSh(S).

Proof

Let A∈PSh(S)A \in PSh(S) and B∈Sep(S)↪PSh(S)B \in Sep(S) \hookrightarrow PSh(S). We need to show that morphisms f:A→Bf : A \to B in PShCPSh_C are in natural bijection with morphisms LsepA→BL_{sep} A \to B in Sep(S)Sep(S).

For ff such a morphism and fU:A(U)→B(U)f_U : A(U) \to B(U) its component over any object U∈SU \in S, consider any covering {Ui→U}\{U_i \to U\}, let S(Ui)→US(U_i) \to U be the corresponding sieve and consider the commuting diagram

obtained from the naturality of PShS(S(Ui)→U,A→fB)PSh_S(S(U_i) \to U, A \stackrel{f}{\to} B).

If for a,a′∈A(U)a,a' \in A(U) two elements that are not equal their restrictions to the cover become equal in that ∀i:a|Ui=a′|Ui\forall i : a|_{U_i} = a'|_{U_i}, then also f(a|Ui)=f(a′|Ui)f(a|_{U_i}) = f(a'|_{U_i}) and since the right vertical morphism is monic there is a uniqueb∈B(U)b \in B(U) mapping to the latter. The commutativity of the diagram then demands that f(a)=f(a′)=bf(a) = f(a') = b.

Since this argument applies to all covers of UU, we have that fUf_U factors uniquely through the projection map A(U)→A(U)/∼U=:Lsep(U)A(U) \to A(U)/\sim_U =: L_{sep}(U) onto the quotient. Since this is true for every object UU we have that ff factors uniquely through A→LsepAA \to L_{sep}A.

Biseparated presheaf

Idea

Often one is interested in separated presheaves with respect to one coverage that are sheaves with respect to another coverage. These are called biseparated presheaves .

of a sheaf topos is given. This is the localization at a set WW of morphisms in Sh(S)Sh(S), with CC the full subcategory of all local objects cc: objects such that Sh(S)(w,c)Sh_(S)(w,c) is an isomorphism for all w∈Ww \in W. A WW-separated object is then called a biseparated presheaf on SS and their collection BiSep(S)BiSep(S) factors the reflective inclusion as